FRM Part I cheat sheet

Sharpe ratio = (E(Rp) − Rf) / σp; it uses total risk, so it is the correct ranking measure only for a fully diversified portfolio.

CAPM: E(Ri) = Rf + βi[E(Rm) − Rf], with βi = Cov(Ri, Rm) / σ²m.

Jensen's alpha = Rp − [Rf + βp(E(Rm) − Rf)]; a positive alpha means the portfolio beat its CAPM benchmark.

Treynor ratio = (Rp − Rf) / βp and uses systematic risk only; the Sortino ratio divides excess return by downside deviation instead of total σ.

The CML plots efficient portfolios against total risk; the SML plots any asset or portfolio against beta, and only the SML prices individual assets.

Two-fund separation: every efficient portfolio is a combination of the risk-free asset and the tangency (market) portfolio.

Diversification drives unsystematic risk toward zero; systematic (market) risk cannot be diversified away and is the only risk earning a premium.

Basel framework three pillars: minimum capital requirements, supervisory review, and market discipline through disclosure.

Agency problems arise when managers pursue perquisites or excess risk; deposit insurance and limited liability create moral hazard.

Two-asset portfolio variance = w1²σ1² + w2²σ2² + 2w1w2ρ12σ1σ2; the cross term vanishes when ρ12 = 0 and is negative when ρ12 < 0.

Bayes: P(A|B) = P(B|A)P(A) / [P(B|A)P(A) + P(B|Aᶜ)P(Aᶜ)].

Standard normal critical values: 90% one-sided 1.645, 95% 1.960, 99% 2.326; a two-sided 95% test uses 1.960.

Standard error of the sample mean = σ/√n; by the central limit theorem the sample mean is approximately normal for large n regardless of population shape.

Sample variance uses divisor n − 1 to be unbiased; population variance uses divisor n.

OLS t-statistic = coefficient / standard error; a large |t| rejects the null that the true coefficient is zero.

Heteroskedasticity leaves OLS coefficients consistent but invalidates standard errors (use White/robust errors); autocorrelation is handled with Newey-West.

Type I error probability = α (significance level); Type II error probability = β; power = 1 − β.

AR(1): y_t = b0 + b1·y_(t−1) + e_t is covariance stationary only if |b1| < 1; b1 = 1 describes a random walk.

Forward price with continuous yield q: F0 = S0·e^{(r−q)T}; for a known cash income, subtract the present value of income from S0 before compounding.

Value of a long forward at time t = (Ft − F0)·e^{−r(T−t)}, where Ft is the forward price quoted at time t for the same maturity.

Futures are marked to market daily through margin accounts; forwards settle only at maturity, so futures carry far less counterparty credit risk.

Put-call parity for European options on a non-dividend stock: c + K·e^{−rT} = p + S0.

A plain vanilla interest rate swap paying fixed and receiving floating equals a long fixed-rate bond financed by a short floating-rate bond.

Covered interest rate parity: F/S = (1 + r_d) / (1 + r_f) for the same maturity, where r_d and r_f are the domestic and foreign rates.

Option Greeks: delta = ∂c/∂S (call 0 to 1, put −1 to 0), gamma = ∂delta/∂S, vega = ∂c/∂σ, theta = ∂c/∂t (usually negative for long options).

Black-Scholes assumes lognormal asset prices, constant volatility and interest rates, no dividends or a known continuous yield, continuous trading and no arbitrage.

A CDS buyer pays a spread and receives the loss given default; the spread approximates annual default probability × (1 − recovery rate).

Parametric VaR for normal returns = |μ − z·σ| over one period; scale by σ·√T for T periods when returns are i.i.d.

Expected shortfall = E[L | L > VaR]; it is coherent, while VaR fails subadditivity and says nothing about the size of losses beyond the quantile.

Modified duration D* = D / (1 + y/m); percentage price change ≈ −D*·Δy + 0.5·Convexity·(Δy)².

DV01 = D* × price × 0.0001; it is the dollar value change for a one basis point move in yield.

Binomial risk-neutral probability p = (e^{rΔt} − d) / (u − d); the option value is the discounted risk-neutral expected payoff.

BSM: c = S0·N(d1) − K·e^{−rT}·N(d2), with d1 = [ln(S0/K) + (r + σ²/2)T] / (σ√T) and d2 = d1 − σ√T; N(d2) is the risk-neutral exercise probability.

EWMA: σ²_t = λ·σ²_(t−1) + (1 − λ)·r²_(t−1); GARCH(1,1) adds a long-run variance term, and its weights must sum to less than one for stationarity.

Merton model: equity is a call on firm assets with strike equal to debt face value, so the risk-neutral default probability is N(−d2).

VaR backtesting under Basel (99%, 250 days): 0–4 exceptions is the green zone, 5–9 yellow, 10 or more red with capital penalties.